PHYSICAL AND MATHEMATICAL SCIENCES. 123

Science, however, was the gainer ; and this theorem,known by the name of “ Cardan’s rule,” at this daymarks a point beyond which no efforts of subsequentalgebraists have been able to carry them in the solutionof cubic equations; and, indeed, but little success hasattended their researches in the general exact solution ofequations of higher degrees. Cardan pointed out the ap-proximation to the roots of those cubic equations whichhis rule did not solve. He also clearly understood someof the leading principles of the roots, and the formationof the co-efficients, when we examine the forms of equa-tions in general synthetically. This must have beenmore difficult than we can readily conceive, owing tothe imperfection of the algebraic language. The usualpractice of the day was to put the rules into verse.Cardan delivered his in a poetical dress. The unavoid-able obscurity of such a mode far more than counter-balances any advantage it may give to the memory.Indeed, even in this respect, a symbolical formula mustfar surpass any versification.

Algebra was at the same time cultivated in Germany .Stiphelius, in a work on that science published at Nu-remburg in 1544, employed integer numeral exponentsof powers, both positive and negative ; and introducedthe same characters for plus and minus which we nowuse. In equations he did not go beyond the second degree.

Robert Recorde , an Englishman, of about the samedate, published die earliest treatise on algebra in the En-glish language, in which the sign of equality first appears.

The properties of algebraic equations were discoveredvery slowly. Pelitarius, a French mathematician, in atreatise bearing the date of 1558, first observed that theroot of an equation is a divisor of the last term. Healso remarked the curious property of numbers, that thesum of the cubes of the natural numbers is the squareof the sum of the numbers themselves.

Bombelli, in Italy , wrote on algebra ; and pointedout that the problems involving the irreducible case ofCardan’s rule, admit of geometrical construction by the